How Does Rotation About a Moving Axis Work in Physics?
There are different types of motions you have studied in the previous classes. One such motion is rotational motion. This is a remarkable type of motion where the direction of a body changes instantaneously and it rotates around an axis. The Rotation about a moving axis is something different. On this concept page, we will learn how a body can move in a rotational fashion around a moving axis. We will also calculate the force, speed, etc along with describing various examples.
What is Rotation?
Apart from the linear motion where the direction of a body does not change, we have studied motion where the direction changes but the speed remains the same. It is a type of motion where the body is moving around an axis at a uniform speed. It cannot be dictated as a velocity as it is changing direction at every instant. Hence, the calculation of force cannot be done when you cannot apply Newton’s Laws of Motion as the body is not moving on a linear path.
Rotation is different from that of revolution. Here, a body is not moving around an axis that does not pass through it. The rotation occurs when the axis passes through the body. The axis passes through the center of mass. If the center of mass is in a translation then the combination of these motions is called rotation about a moving axis.
What is Rotational Motion?
As we have discussed what rotation is, it becomes a lot easier to understand what rotational motion is. In this case, we will use a few rotational motion examples to make the discussion better.
Consider a rigid non-uniform body with a center of mass. If an axis passes through the center of mass and the body rotates on it, it is called rotational motion. When a top is spun and rotates stagnantly on the floor, it is called rotational motion. Here, you will find that the body is not translating on the floor. There is no linear motion. The only motion it has is the rotational motion. If you lift your bicycle and pedal, you will find only the back wheel is rotating but the cycle is not moving. It rotates around the axle where it is fixed. It is also an example of rotational motion.
Rotational Motion about a Moving Axis
Proceeding to the next level, when a rotating body is also translating, it is called rotation around a moving axis. If you have understood the concepts about rotation then consider the axis of this moving body is translating linearly. Let us consider the same example. When a top is spinning on its axis but the axis is also moving on a smooth floor. It is called rotational motion on a moving axis. This new concept gives birth to a whole new segment of physics called rotational mechanics.
How Can You Calculate Rotational Force and Torque?
In this case, we will have to understand and calculate two different momentums. The body of mass M will have a linear momentum for the translation and a rotational force. Hence,
Linear Momentum p = MVc
(where Vc = linear velocity of the center of mass)
If there is any change in the momentum, a force has been applied as per Newton’s Second Law of Motion. Hence,
F = dp/dt
(where F = net force applied on the body)
Here comes the part where you have to calculate the angular momentum. Considering a reference point, the angular momentum of a body can be expressed as,
L = Lc + (r x p)
(where, Lc represents the angular momentum, r is the vector calculated from the reference point to the center of mass and p is the linear momentum)
You can see how the angular momentum is connected with the linear momentum as all the mass present in that body is attached to the center of mass. (r x p) is considered to be the orbital angular momentum. If there is any change in the angular momentum, it can be represented by the following equation.
𝜏 = dL/dt
The kinetic energy of this body M can be calculated as
K = ½(MVc2 + Iω2)
This is how rotational force is calculated. Follow this section to understand how the torque calculation for the rotating body is done too.
Example of Rotational Motion around a Moving Axis
You have now understood how to calculate and determine the rotational force of a rotating body. Let us consider a few examples to understand what this type of motion is and how it differs from rotational motion.
When the earth rotates on its axis, it is called rotational motion. While doing this, its axis is also under translation. It is called rotational motion around a moving axis.
When you are spinning a key ring in your finger and walking, the key ring’s axis is witnessing rotational motion around itself along with a translation due to your walking.
When a drum rolls down an inclined plane. It witnesses rotating motion and its axis moves linearly.
FAQs on Rotation About a Moving Axis: Key Concepts & Applications
1. What is meant by rotation about a moving axis in Physics?
Rotation about a moving axis describes the motion of a rigid body that is simultaneously rotating about an axis and translating (moving linearly) through space. This combined motion, often called general plane motion, can be analysed by considering the translational motion of the body's centre of mass and the rotational motion about an axis passing through this centre of mass. A common example is a car wheel rolling on a road.
2. How does rotation about a moving axis differ from pure rotational motion?
The primary difference lies in the movement of the axis of rotation itself.
- In pure rotational motion, the object spins around a stationary or fixed axis. For instance, a ceiling fan's blades rotate, but the central hub does not move through the room.
- In rotation about a moving axis, the object rotates around an axis that is itself moving. The wheel of a moving bicycle is a perfect example; it rotates about its axle, while the axle translates forward with the bicycle.
3. What are some common real-world examples of rotation about a moving axis?
This type of combined motion is frequently observed in everyday life. Some key examples include:
- A bowling ball rolling down a lane.
- The wheels of a car or a bicycle moving along a road.
- A cylinder or sphere rolling down an inclined plane.
- The planet Earth orbiting the Sun, where Earth rotates on its own axis while that axis simultaneously translates along its orbital path.
4. Why is the concept of the centre of mass so important for analysing this type of motion?
The centre of mass is a crucial concept because it allows us to simplify a complex motion into two separate, manageable parts. As per the CBSE syllabus for 2025-26, we can analyse the motion as:
- The translational motion of the entire body, treated as if all its mass were concentrated at the centre of mass.
- The rotational motion of the body about an axis passing through its centre of mass.
5. How is the total kinetic energy of a body in combined translational and rotational motion calculated?
The total kinetic energy (K) of a body rotating about a moving axis is the sum of its translational kinetic energy and its rotational kinetic energy. The formula is:
K = Ktranslational + Krotational = ½MVcm² + ½Icmω²
Where:
- M is the total mass of the body.
- Vcm is the linear velocity of the centre of mass.
- Icm is the moment of inertia about the axis through the centre of mass.
- ω is the angular velocity of the rotation.
6. What is the condition for a wheel 'rolling without slipping', and what is its significance?
For an object of radius R to roll without slipping, the linear velocity of its centre of mass (Vcm) must be related to its angular velocity (ω) by the formula:
Vcm = Rω
The significance of this condition is that the point of the object in contact with the surface is momentarily at rest. This means there is no sliding or skidding, and static friction may act, but no kinetic friction is involved, preventing energy loss due to friction.
7. How does torque affect an object rotating about a moving axis?
Torque's role can be understood by separating the motion. An external net force applied to the centre of mass changes the body's translational motion (its linear velocity, Vcm). An external net torque about the centre of mass changes the body's rotational motion (its angular velocity, ω). For example, applying brakes to a car wheel creates a torque to slow its rotation, while the force of friction from the road slows its translation.
8. How does the law of conservation of energy apply to a cylinder rolling down an incline?
When a cylinder rolls down an inclined plane without slipping, its initial gravitational potential energy is converted into kinetic energy. However, this kinetic energy has two forms: translational kinetic energy (due to the movement of its centre of mass) and rotational kinetic energy (due to its spinning). Therefore, the loss in potential energy (mgh) equals the gain in total kinetic energy (½MVcm² + ½Icmω²), demonstrating the conservation of mechanical energy.
9. Can an object rotating about a moving axis have zero angular momentum?
Yes, it's theoretically possible under specific conditions. The total angular momentum is the vector sum of the orbital angular momentum (due to the motion of the centre of mass) and the spin angular momentum (due to rotation about the centre of mass). These two vectors could be equal in magnitude and opposite in direction, resulting in a net angular momentum of zero, even though the object is in motion.
10. If two cylinders of the same mass and radius, one solid and one hollow, roll down an incline, which one reaches the bottom first?
The solid cylinder will reach the bottom first. This is because the hollow cylinder has a larger moment of inertia (I), as its mass is distributed farther from the axis of rotation. A larger moment of inertia means more energy is required to make it rotate. Therefore, for the same amount of potential energy converted, the hollow cylinder will have less translational kinetic energy and thus move slower down the incline compared to the solid cylinder.
